Theorem ax17m 218

Description: Axiom to quantify a variable over a formula in which it does not occur. Axiom C5 in [Megill] p. 444 (p. 11 of the preprint). Also appears as Axiom B6 (p. 75) of system S2 of [Tarski] p. 77 and Axiom C5-1 of [Monk2] p. 113. (Contributed by Mario Carneiro, 10-Oct-2014.)

Hypothesis

Ref	Expression
ax17m.1	|- A:*

Assertion

Ref	Expression
ax17m	|- T. |= [A ==> (A.\x:al A)]

Distinct variable groups:   x,A   al,x

Proof of Theorem ax17m

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax17m.1	. 2 |- A:*
2		wv 64	. . 3 |- y:al:al
3	1, 2	ax-17 105	. 2 |- T. |= [(\x:al Ay:al) = A]
4	1, 3	isfree 188	1 |- T. |= [A ==> (A.\x:al A)]

Syntax hints:  tv 1  *hb 3  kc 5  \kl 6  T.kt 8  [kbr 9   |= wffMMJ2 11  wffMMJ2t 12   ==> tim 121  A.tal 122

This theorem was proved from axioms:  ax-syl 15  ax-jca 17  ax-simpl 20  ax-simpr 21  ax-id 24  ax-trud 26  ax-cb1 29  ax-cb2 30  ax-wctl 31  ax-wctr 32  ax-weq 40  ax-refl 42  ax-eqmp 45  ax-ded 46  ax-wct 47  ax-wc 49  ax-ceq 51  ax-wv 63  ax-wl 65  ax-beta 67  ax-distrc 68  ax-leq 69  ax-distrl 70  ax-wov 71  ax-eqtypi 77  ax-eqtypri 80  ax-hbl1 103  ax-17 105  ax-inst 113  ax-eta 177

This theorem depends on definitions:  df-ov 73  df-al 126  df-an 128  df-im 129

This theorem is referenced by: (None)